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Division of polynomials

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Yann Villarreal
Yann Villarreal

This document discusses algebraic division of polynomials. It begins by explaining the laws of exponents used in polynomial division. Specifically, it covers dividing powers of the same base and the meaning of the zero exponent. It then describes the three types of algebraic division: division of monomials, division of a polynomial by a monomial, and division of two polynomials. Steps for performing each type of division are provided. An example problem demonstrates dividing two polynomials.

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ALGEBRAIC DIVISION BY: MNICA ELIZONDO YANN VILLARREAL
OBJECTIVES By the end of this lesson you should be able to: Perfom divisions of polynomials
LETS GET STARTED! To perform the operation of division you have to apply certain laws of the exponents that we havent seen yet, but dont worry! because we are going to proceed to explain them in the following slides.
LAW OF EXPONENTS RELATIVE TO THE DIVISION OF POWERS OF EQUAL BASE Imagine we want to divide by , where is a real number and not null (which means not equal to zero), and are natural numbers and also , then we can write this operation as . In the numerator we have the factor repeated times, whereas in the denominator the factor is present times. Since we have the same factor repeated in the numerator and denominator we can cancel out the same amount of factors that are in both parts of the division. Therefore we have that = .
ZERO EXPONENT Now that we are considering exponents you might ask what is the meaning of powering a factor to the zero exponent? In order to explain this lets use the previous rule of powers of the division. Be a real and not null number and a natural number different from zero. Let us obtain by two different ways. The first is by applying the rule of division of powers, in this case we have that = 0 . Also we know that = 1, because both factors cancel each other. So we conclude that for all 0, 0 = = = 1, this is 0 = 1.
ALGEBRAIC DIVISION Do you remember from the previous topic that there are different types of algebraic multiplications? Well the same applies for algebraic division. There are three different types of operation that can be done algebraically and they are: 1. Division of monomials 2. Division of a polynomial by a monomial 3. Division of two polynomials
DIVISION OF MONOMIALS To perform divisions of two monomials just follow the next steps: 1. The sign of the quotient of the division is determined by the rule of signs, which states that if two numbers of equal signs are divided the quotient is positive, whereas if they have contrary signs the quotient must be negative. 2. Then we divide the coefficient of the numerator by the coefficient of the denominator. 3. Finally we divide the literals (the part of the monomials that is composed of variables) by using the next properties of the exponents for the division: = > 1 = 1 > Where is a real number and not equal to zero.
PROBLEMS Solve the following divisions of monomials: a) 88 12 25 7 b) 243 12 9 68 9 c) 304 2 5ヰ
ANSWERS a) 88 12 25 7 = 43 5 b) 243 12 9 68 9 = 42 4 c) 304 2 5ヰ = 63
DIVISION OF POLYNOMIAL BY A MONOMIAL To perform divisions of a polynomial you just have to divide each of the terms of the polynomial by the monomial. Lets see an example: Perform the division of 165122+4 4 Solution: 44 3 + 1 165122+4 4 = 165 4 122 4 + 4 4 = 44 3 + 1
DIVISION OF TWO POLYNOMIALS In general, to divide two polynomials follow the next steps: 1. The dividend and divisor are ordered according to the decreasing exponents of one variable that appears in both, including terms with coefficient zero for the missing powers. 2. It is divided the first term of the dividend by the first term of the divisor to obtain the first term of the quotient. 3. The first term of the quotient is multiplied by the divisor and it is subtracted the product of the dividend, the difference that is obtained is the new dividend. 4. To find the second term and all the consecutives of the quotient, the previous steps are repeated until the degree of the polynomial obtained is lees than the degree of the divisor. 5. The results is then verified with the formula: ＄ ｐ + = ｐ We recommend you to prove the result of all your divisions, jut to make sure that you followed all steps correctly.
EXAMPLE Solve the following division: (63 + 52 9 7)/(2 + 1) Procedure 1. Divide 63 歎 2 2. Multiply 32 by 2x + 1 and subtract the product of the dividend and reduce like terms. 3. Divide 22 歎 2 4. Multiply x by 2 + 1 and subtract the product of the new dividend 5. Divide 10 歎 2 6. Multiply 5 by 2 + 1 and subtract from the new dividend 7. Because the difference (2) is of less degree than the divisor since (2) is constant and, therefore, the degree of the term is zero, here finishes the operation. Solution: In summary, the quotient is 32 + 5 and the reminder is 2.

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Division of polynomials

  • 1. ALGEBRAIC DIVISION BY: MNICA ELIZONDO YANN VILLARREAL
  • 2. OBJECTIVES By the end of this lesson you should be able to: Perfom divisions of polynomials
  • 3. LETS GET STARTED! To perform the operation of division you have to apply certain laws of the exponents that we havent seen yet, but dont worry! because we are going to proceed to explain them in the following slides.
  • 4. LAW OF EXPONENTS RELATIVE TO THE DIVISION OF POWERS OF EQUAL BASE Imagine we want to divide by , where is a real number and not null (which means not equal to zero), and are natural numbers and also , then we can write this operation as . In the numerator we have the factor repeated times, whereas in the denominator the factor is present times. Since we have the same factor repeated in the numerator and denominator we can cancel out the same amount of factors that are in both parts of the division. Therefore we have that = .
  • 5. ZERO EXPONENT Now that we are considering exponents you might ask what is the meaning of powering a factor to the zero exponent? In order to explain this lets use the previous rule of powers of the division. Be a real and not null number and a natural number different from zero. Let us obtain by two different ways. The first is by applying the rule of division of powers, in this case we have that = 0 . Also we know that = 1, because both factors cancel each other. So we conclude that for all 0, 0 = = = 1, this is 0 = 1.
  • 6. ALGEBRAIC DIVISION Do you remember from the previous topic that there are different types of algebraic multiplications? Well the same applies for algebraic division. There are three different types of operation that can be done algebraically and they are: 1. Division of monomials 2. Division of a polynomial by a monomial 3. Division of two polynomials
  • 7. DIVISION OF MONOMIALS To perform divisions of two monomials just follow the next steps: 1. The sign of the quotient of the division is determined by the rule of signs, which states that if two numbers of equal signs are divided the quotient is positive, whereas if they have contrary signs the quotient must be negative. 2. Then we divide the coefficient of the numerator by the coefficient of the denominator. 3. Finally we divide the literals (the part of the monomials that is composed of variables) by using the next properties of the exponents for the division: = > 1 = 1 > Where is a real number and not equal to zero.
  • 8. PROBLEMS Solve the following divisions of monomials: a) 88 12 25 7 b) 243 12 9 68 9 c) 304 2 5ヰ
  • 9. ANSWERS a) 88 12 25 7 = 43 5 b) 243 12 9 68 9 = 42 4 c) 304 2 5ヰ = 63
  • 10. DIVISION OF POLYNOMIAL BY A MONOMIAL To perform divisions of a polynomial you just have to divide each of the terms of the polynomial by the monomial. Lets see an example: Perform the division of 165122+4 4 Solution: 44 3 + 1 165122+4 4 = 165 4 122 4 + 4 4 = 44 3 + 1
  • 11. DIVISION OF TWO POLYNOMIALS In general, to divide two polynomials follow the next steps: 1. The dividend and divisor are ordered according to the decreasing exponents of one variable that appears in both, including terms with coefficient zero for the missing powers. 2. It is divided the first term of the dividend by the first term of the divisor to obtain the first term of the quotient. 3. The first term of the quotient is multiplied by the divisor and it is subtracted the product of the dividend, the difference that is obtained is the new dividend. 4. To find the second term and all the consecutives of the quotient, the previous steps are repeated until the degree of the polynomial obtained is lees than the degree of the divisor. 5. The results is then verified with the formula: ＄ ｐ + = ｐ We recommend you to prove the result of all your divisions, jut to make sure that you followed all steps correctly.
  • 12. EXAMPLE Solve the following division: (63 + 52 9 7)/(2 + 1) Procedure 1. Divide 63 歎 2 2. Multiply 32 by 2x + 1 and subtract the product of the dividend and reduce like terms. 3. Divide 22 歎 2 4. Multiply x by 2 + 1 and subtract the product of the new dividend 5. Divide 10 歎 2 6. Multiply 5 by 2 + 1 and subtract from the new dividend 7. Because the difference (2) is of less degree than the divisor since (2) is constant and, therefore, the degree of the term is zero, here finishes the operation. Solution: In summary, the quotient is 32 + 5 and the reminder is 2.